Use a loudspeaker
array to reproduced a desired wave field can be seen as a space sampling of
the field itself. Just as the time sampling, the space sampling is governed
by the Nyquist theorem, which in this case is only a litle more difficult.
Imagine a plane
wave inpinging on a discrete speakes array: the speaker spacing is
fundamental for a correct sampling; Nyquist says in this case tht the
projection of the wavelength on the array must at least be twice the speaker
spacing. So this is the formula coming out. Above a certain frequency, the
wave must be limited within a certain incidence angle.
Just like in the
time domain case, besides a sampling contitions, a reconstruction condition
must be respected. It imposes a specific directivity on secondary sources. In
pratice if for a given frequency I see a maximum incidence angle, that
frequency must not be delivered by secondary sources at angle wider than the
limit one.
Take for example
the limit case of a plane waves frontally impacting the array: phi is zero so
the sampling condition is always verified. But i still need a reconstruction
filter, because the secondary fronts are circular, and only in a suffuciently
far field their envelope will be flat, whilst in the near field i will
experience the spatial aliasing. It’s intuitive to understand that rigid
pistons instead of point sources con solve the problem, since the front is
flat from the beginning. But we know that a rigid pistons is nothing else
than a directional sources, with beamwidth that decreases with frequency. It
is not exacly the directivity requested by the formula, because we have lobes
for example, but qualitativly it is according with it.
